Adaptive backstepping cancelation of unmatched unknown sinusoidal disturbances for unknown LTI systems by state derivative feedback

Solutions already exist for the problem of canceling sinusoidal disturbances by measurement of the state or by measurement of an output for unknown linear and nonlinear systems. In this paper, we design an adaptive backstepping controller to cancel sinusoidal disturbances forcing an unknown linear time-invariant system in controllable canonical form which is augmented by a linear input subsystem with unknown system parameters by using only measurement of state-derivatives of the original subsystem and state of the input subsystem. Our design is based on four steps, (1) parametrization of the sinusoidal disturbance as the output of a known feedback system with an unknown output vector that depends on both unknown disturbance parameters and unknown plant parameters, (2) design of an adaptive disturbance observer for both disturbance and its derivative, (3) design of an adaptive controller for virtual control input, and (4) design final adaptive controller by using backstepping procedure. We prove that the equilibrium of the closed-loop adaptive system is stable and state of the considered original subsystem goes to zero as t → ∞ with perfect disturbance estimation. The effectiveness of the controller is illustrated with a simulation example of a third order system.